We use a new gyrokinetic threshold model to predict a bifurcation in tokamak pedestal width-height scalings that depends strongly on plasma shaping and aspect-ratio. The bifurcation arises from the first and second stability properties of kinetic-ballooning-modes that yields wide and narrow pedestal branches, expanding the space of accessible pedestal widths and heights. The wide branch offers potential for edge-localized-mode-free pedestals with high core pressure. For negative triangularity, low-aspect-ratio configurations are predicted to give steeper pedestals than conventional-aspect-ratio. Both wide and narrow branches have been attained in tokamak experiments.

The realization of magnetic confinement fusion energy represents a significant milestone in the quest for clean and abundant power sources. This endeavor hinges on the ability to confine a high-pressure plasma with magnetic fields, as characterized by the ratio $ \beta = 2 \mu 0 p / B 2$, where *p* is the plasma pressure and *B* the magnetic field strength. For tokamaks operating in the high-confinement mode (H-mode), an intriguing phenomenon occurs—the formation of a pedestal at the plasma edge,^{1} often enhancing *β* substantially and hence fusion power. However, ballooning modes pose a fundamental challenge to achieving optimal *β* values. Recent experiments in negative triangularity plasmas^{2,3} showed that plasma shaping changes pedestal ballooning stability substantially. In this Letter, we describe a bifurcation in the H-mode pedestal width and height that can be manipulated and optimized with plasma shaping and aspect-ratio. The bifurcation arises from ballooning stability properties and presents new pedestal operating scenarios for reactors. By degrading ballooning stability in the edge, higher core *β* may be achieved by enabling the pedestal to grow without triggering edge-localized-modes (ELMs) that threaten plasma-facing components.^{4,5}

Predictive models constrain the pedestal radial width $ \Delta ped$ and height $ \beta \theta , ped$ by ballooning and ELM stability.^{5,6} Microscopic ballooning stability gives a width-height scaling that constrains pedestal pressure gradients. The macroscopic ELM constraint gives a “hard” limit that transports significant pressure once triggered.^{7,8} These two constraints intersect to give a $ \Delta ped , \u2009 \beta \theta , ped$ prediction. Previous work has shown that the ELM constraint is sensitive to plasma shaping: negative triangularity TCV tokamak plasmas were subject to a much more restrictive ELM constraint than positive triangularity,^{9} whereas on the DIII-D tokamak, fueling and strong positive triangularity provided a route to “Super H-mode”^{10,11} pedestals. The new bifurcation in this work complements these results, focusing on kinetic-ballooning-mode (KBM)^{12} physics.

^{5,13}so that pedestal structure can be varied meaningfully with two variables,

*ψ*at the pedestal top, and $ B \xaf pol = \mu 0 I p / l$ with last-closed-flux-surface circumference

*l.*

^{5,14}The width-height scaling is the boundary in $ \Delta ped , \u2009 \beta \theta , ped$ space separating pedestals that are limited by KBMs. For more details, see Refs. 5 and 15. Experimentally, $ \Delta ped$ and $ \beta \theta , ped$ typically grow close to the width-height scaling trajectory,

^{16}facilitated by a radially broadening flow shear profile that suppresses long-wavelength turbulence. For ELMy H-modes, growth ceases when peeling-ballooning-modes (PBMs) are destabilized, causing an ELM.

^{5,17–19}

^{20}At relatively low pressure gradients, the normalized gradient,

^{21}Here,

*V*is the flux-surface enclosed volume and

*R*

_{0}the plasma major radius. At much higher

*α*, the local magnetic shear at the low-field side becomes highly negative and stabilizing, which leads to second stability.

^{20}A key result of this work is that first and second KBM stability across the pedestal is realized as two distinct branches—radially wide and narrow—of ballooning width-height scalings. For ELMy H-modes, given that the ELM limit is related to stored pedestal energy,

^{22}lower gradient pedestals are intrinsically “wider” and steeper gradient pedestals are “narrower.” We will also show that wide pedestals offer potential for robust ELM-free operation.

In Fig. 1, we show the pedestal width-height bifurcation for National-Spherical-Torus-Experiment (NSTX) discharges 129015, 129038, 132543, 139034, and 139047.^{23,24} The width-height scaling lines in Fig. 1 are obtained with a linear KBM threshold model^{25} using GS2 gyrokinetic simulations.^{26} Each NSTX discharge admitted two solutions: a wide and narrow branch. However, 139034, 139047, and 129038 equilibria—indicated by markers in Fig. 1—lie on the wide branch, whereas 129015 and 132543 lie on the narrow branch. These discharges demonstrate that NSTX experiments occupy both wide and narrow pedestal branches. For readability, in Fig. 1 we only plot branches closest to the experimental point. We also plot equilibrium points and width-height scalings for DIII-D 163303,^{27} lying on the narrow branch, MAST 29782,^{14} also on the narrow branch, and a case representing the Primary Reference Discharge for SPARC,^{28–30} whose operational point is projected below the narrow branch. While we do not calculate a scaling, we show the ELMy H-mode pedestal C-Mod 1101214029 equilibrium point^{31,32} lies in the narrow branch region.

We perform ideal $ s \u2212 \alpha $ analysis^{20} in Fig. 2 at the pedestal mid-radius for equilibria on the wide and narrow branches of NSTX 139047, shown by A, B and C, D in Fig. 1 (we omit the 139047 narrow branch scaling in Fig. 1). We plot ideal stability curves vs *α* and magnetic shear $ s = ( r / q ) ( d q / d r )$, where *r* is the flux-surface half-diameter. The wide branch points A and B are in the ideal first-stable region, and the narrow branch points C and D are ideal second-stable below the stability curve “nose.” Because the scalings in Fig. 1 use gyrokinetic stability, points A–D are relatively far from the ideal boundaries.

We now demonstrate how plasma shaping and aspect-ratio enter the ballooning bifurcation. The geometric shape of the last-closed-flux-surface and shaping's penetration to the magnetic axis impacts plasma performance^{2,33–35} and changes ballooning mode stability.^{36,37} Starting from an experimental equilibrium for NSTX 139047, we vary the shaping parameters and aspect-ratio. We use Luce parameters for the plasma shape,^{38} defining the elongation *κ* and triangularity *δ* as the average $ \u27e8 \cdots \u27e9 L$ of the Luce parameters $ \kappa = \u27e8 \kappa \u27e9 L , \u2009 \u2009 \delta = \u27e8 \delta \u27e9 L$. When varying *δ*, the total plasma current *I _{p}* is held constant. Additionally, due to its impact on ballooning stability,

^{39}the quantity $ \beta N = 2 \mu 0 \u27e8 p \u27e9 a / I p B T 0$ is held constant. Here, $ \u27e8 p \u27e9$ is the volume-averaged pressure,

*a*the minor radius, and $ B T 0$ the toroidal magnetic field strength at the magnetic axis. When varying

*κ*, we scale $ I p \u223c 1 + \kappa 2$, and when varying aspect-ratio $ A = R 0 / a$, we scale $ \beta N \u223c 1 / R 0 , \u2009 \u2009 B T 0 \u223c R 0 , I p \u223c R 0$.

^{40}

We first describe the effect of shaping and aspect-ratio on ideal-ballooning-mode (IBM) stability (note that ideal and kinetic ballooning stability boundaries can differ, often significantly^{12,25,41–45}). For NSTX 139047, we compare equilibria with different *κ*, *δ*, and *A* values with ideal $ s \u2212 \alpha $ analysis. In Fig. 3(a), increasing elongation from $ \kappa = 1.4$ to 2.3 shifts the first-stable boundary to much higher *s* and *α*. This suggests that a KBM first-stable width-height scaling branch will be much wider radially for a similar pedestal height (i.e., larger $ \Delta ped$) at lower elongation because the first-stable boundary (in $ s \u2212 \alpha $ space) is at lower *α* values. Similarly, in Fig. 3(b), decreasing triangularity to negative values decreases the *s* and *α* values for the first-stable boundary, suggesting that a KBM first-stable, wide branch will be much wider at lower and negative triangularity. Finally, Fig. 3(c) suggests that the first-stable branch will be widest at high-aspect-ratio.

*s*and

*α*. As predicted from $ s \u2212 \alpha $ analysis in Fig. 3, the wide branch in Fig. 4(a) widens as triangularity becomes more negative, whereas for the narrow branch, larger positive triangularity increases the width. The wide and narrow branches scalings for the KBM are

^{5,46}and hence, bifurcations may not always be predicted with ideal MHD.

In the right column of Fig. 5, we show scalings for a range of $ \beta \theta , ped$ values. Curiously, at conventional-aspect-ratio $ \u2273 2.5$, the wide branch has gentle gradients close to L-mode-like values [ $ \u2212 a \u2207 \u2009 ln ( T e ) \u226a 10$], which may explain why no conventional-aspect-ratio H-mode experiments we have studied were in the wide-branch, despite finding wide-branch solutions in such equilibria. Notably, elongated, low-aspect-ratio plasmas, such as in NSTX, have a smaller gap in $ \beta \theta , ped$ between the narrow and wide branches, which might explain why they are both are accessible for NSTX, shown in Fig. 1.

^{20}

^{,}

^{47}In Fig. 6, we also plot the in-flux-surface magnetic curvature drift frequency,

*κ*,

*δ*, and

*A*values but the same $ \Delta ped , \u2009 \beta \theta , ped$ values. The dimensionless frequency $ \omega \kappa $ is normalized to the ion parallel streaming frequency using the quantity

*G*. For ballooning modes, weaker magnetic shear and faster magnetic drifts tend to stabilize the modes. Crucially, local shear stabilization is sign-independent.

^{20}Because ballooning eigenmodes peak in magnitude close to the low-field-side with poloidal angle $ \theta \u2248 0$, the values of $ s local$ and $ \omega \kappa $ around $ \theta \u2248 0$ most strongly determine ballooning mode stability. Figure 6 shows that lower values of elongation, more negative triangularity, and higher aspect-ratio decrease the local shear and increase $ \omega \kappa $, all of which destabilize the KBM. This is consistent with Fig. 3 where, for example, in Fig. 3(a), relatively small values of

*α*are needed to destabilize the ballooning mode at lower elongation, which has relatively small $ s local$ around

*θ*= 0 and very fast magnetic drifts. The varying impact of kinetic effects, such as drift resonances and Landau damping across shaping and aspect-ratio, while included in our gyrokinetic simulations, has not been examined closely here, but is an important question for future work.

We have discovered that the width-height scaling (a) has a bifurcation and (b) has a strong dependence on shaping and aspect-ratio. We now employ these findings to predict favorable ELM-free regimes.

Degraded pedestal gradients might permit ELM-free operation while achieving a high pedestal pressure,^{4} which could be achieved in the wide KBM branch. In Fig. 7, we schematically show the wide and narrow KBM branches, with color contours heuristically representing different shaping and aspect-ratio configurations based on results in Figs. 4 and 5. Some possible equilibrium points are shown in Fig. 7; point (a): a conventional-aspect-ratio ELMy H-mode pedestal typically saturates where first ELM stability, curve *E*_{1}, intersects the narrow branch at (a). We have assumed that the ELM constraint has the scaling dependence $ \Delta ped \u223c \beta \theta , ped 4 / 3$.^{6} Point (b): Super H-modes^{11} are obtained by shifting the ELM stability boundary to *E*_{2}, resulting in a much higher pedestal at (b), but possibly still ELMing.

It may be possible to move to a much higher $ \beta \theta , ped$ value and avoid ELMs by accessing the wide pedestal branch. In Fig. 7, we plot a new constraint *F* resulting from additional transport, flow shear degradation, or other saturation mechanisms. $ F ( \Delta ped , \beta \theta , ped )$ will vary by saturation mechanism, but preliminary analysis shows $ \Delta ped \u223c \beta \theta , ped \u2212 1$ for a flow shear degradation constraint.^{15} The intersection of the wide branch and *F* could give a much higher and wider ELM-free pedestal at point (c). However, ELMy solutions may still exist on the wide branch: if an even higher $ \beta \theta , ped$ is desired, shaping and aspect-ratio could be adjusted to move to points (d) or (e). For this illustration, we have omitted how plasma shaping and aspect-ratio changes ELM stability.^{9,10} Determining the dependence of *both* the ELM and KBM scalings on shaping and aspect-ratio, and examining the effects of shaping and aspect-ratio with other ELM control techniques^{16} is an important problem.

To test whether narrow and wide KBM branches are likely to ELM, we also perform ideal PBM simulations for NSTX 139047 using ELITE.^{7,48,49} We find that the plasma is PBM-stable along wide and narrow KBM branches for mode numbers $ n = 3 \u2212 25$, approaching instability for very large $ \Delta ped$ or $ \beta \theta , ped$ values. An important caveat is that non-ideal effects can modify PBM stability in NSTX.^{50}

At low-aspect-ratio, negative triangularity (NT) might be particularly attractive for ELM-free H-mode operation. NT^{2} has received attention for its ELM-free and enhanced confinement characteristics.^{3,51} By design, NT does not access H-mode, staying in first ballooning stability and L-mode^{37} to avoid ELMs. While conventional-aspect-ratio NT degrades first-stable pedestals so much that edge gradients are L-mode-like,^{3} at *low-aspect-ratio* NT may achieve steeper gradients: Fig. 5(c) shows the wide branch has roughly twice as steep gradients at *A* = 1.7 compared with *A* = 2.8 due to $ \Delta ped , wide \u223c A 1.5$ [see Eq. (4)]. Thus, a low-aspect-ratio NT H-mode could inherit the favorable properties of NT (no narrow branch access) with H-mode-like gradients.

While we lack a prescription for accessing separate branches of the width-height scaling bifurcation, there is a curious observation: a sufficiently big and sudden loss (or gain) of pedestal particles and heat—such as an ELM (or pellet injection)—could cause a pedestal to jump from the narrow to wide branch (or wide to narrow branch) in Fig. 7. There are examples of H-modes that become ELM-free after an initial ELM;^{52} jumping from narrow to wide branches is a possible explanation. Finally, companion NSTX discharges 129015 and 129038,^{23} shown in Fig. 1, fall in the narrow and wide branches. These discharges differ mainly by lithium dosing,^{53} which may be responsible for different branch access. Two other largely ELM-free regimes, I-mode^{54} and wide pedestal QH-mode,^{55} may also access the wide pedestal branch.

We have discovered a novel and intriguing bifurcation in the width-height scalings of tokamak pedestals, which can be modified by plasma shaping and aspect-ratio.^{33} This bifurcation arises from the first and second stability properties of kinetic-ballooning-modes that give distinct wide and narrow pedestal branches. This discovery opens up the operating space for accessible pedestal widths and heights, offering new prospects for pedestal regimes in fusion experiments. While our investigations thus far have only revealed wide branch access in NSTX pedestals, investigation of how NSTX pedestals access the wide branch could provide a pathway to ELM-free operation^{3} with high pedestal pressures.

We are grateful to S. C. Cowley, W. Dorland, R. Maingi, O. Sauter, P. B. Snyder, and H. R. Wilson for insightful discussions and to T. Bechtel and J. McClenaghan for technical assistance with the EFUND code. This work was supported by the U.S. Department of Energy under Contract Nos. DE-AC02-09CH11466, DE-SC0022270, DE-SC0022272, DE-SC0014264, and DE-SC0021629, and the Department of Energy Early Career Research Program. The United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purpose.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**J. F. Parisi:** Conceptualization (lead); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **A. O. Nelson:** Conceptualization (lead); Data curation (equal); Investigation (lead); Methodology (lead); Software (lead); Validation (equal); Writing – review & editing (equal). **R. Gaur:** Conceptualization (equal); Formal analysis (lead); Software (lead); Validation (lead). **S. M. Kaye:** Conceptualization (equal); Funding acquisition (lead); Supervision (lead); Writing – review & editing (lead). **F. I. Parra:** Conceptualization (equal); Investigation (equal); Supervision (lead). **J. W. Berkery:** Conceptualization (equal); Data curation (lead); Methodology (equal); Writing – review & editing (equal). **K. Barada:** Formal analysis (equal). **C. Clauser:** Investigation (equal); Software (equal). **A. J. Creely:** Data curation (equal). **A. O. Diallo:** Data curation (lead); Methodology (equal); Supervision (supporting). **W. Guttenfelder:** Conceptualization (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (supporting). **J. W. Hughes:** Data curation (equal); Methodology (supporting). **L. A. Kogan:** Data curation (equal). **A. Kleiner:** Conceptualization (supporting); Data curation (equal); Formal analysis (equal); Methodology (equal); Validation (equal). **A. Q. Kuang:** Data curation (equal). **M. Lampert:** Data curation (lead). **T. Macwan:** Formal analysis (equal). **J. E. Menard:** Data curation (supporting); Methodology (equal); Supervision (supporting). **M. A. Miller:** Data curation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Princeton Data Commons at https://doi.org/10.34770/fc81-3051, Ref. 56. Part of the data analysis was performed using the OMFIT integrated modeling framework^{57} using the Github project gk_ped.^{15}